Sébastien Loisel
The $p$-Laplacian is a nonlinear partial differential equation, parametrized by $p \in [1,\infty]$. We provide new numerical algorithms, based on the barrier method, for solving the $p$-Laplacian numerically in $O(\sqrt{n}\log n)$ Newton iterations for all $p \in [1,\infty]$, where $n$ is the number of grid points. We confirm our estimates with numerical experiments.
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In this paper, we study a posteriori error estimators which aid multilevel iterative solvers for linear systems with graph Laplacians. In earlier works such estimates were computed by solving global optimization problems, which could be computationally expensive. We propose a novel strategy to compute these estimates by constructing a Helmholtz decomposition on the graph based on a spanning tree and the corresponding cycle space. To compute the error estimator, we solve efficiently the linear system on the spanning tree, and then we solve approximately a least-squares problem on the cycle spac…
Shifted Laplacian based multigrid preconditioners for solving indefinite Helmholtz equations
本文提出结合移位拉普拉斯和原方程的新型预处理器，通过加速Krylov迭代并改进多网格方法求解不定亥姆霍兹方程。
In this paper we propose and analyze new efficient sparse approximate inverse (SPAI) smoothers for solving the two-dimensional (2D) and three-dimensional (3D) Laplacian linear system with geometric multigrid methods. Local Fourier analysis shows that our proposed SPAI smoother for 2D achieves a much smaller smoothing factor than the state-of-the-art SPAI smoother studied in [Bolten, M., Huckle, T.K. and Kravvaritis, C.D., 2016. Sparse matrix approximations for multigrid methods. Linear Algebra and its Applications, 502, pp.58-76.]. The proposed SPAI smoother for 3D cases provides smaller optim…
In this paper, we develop a Monte Carlo method for solving PDEs involving an integral fractional Laplacian (IFL) in multiple dimensions. We first construct a new Feynman-Kac representation based on the Green function for the fractional Laplacian operator on the unit ball in arbitrary dimensions. Inspired by the "walk-on-spheres" algorithm proposed in [24], we extend our algorithm for solving fractional PDEs in the complex domain. Then, we can compute the expectation of a multi-dimensional random variable with a known density function to obtain the numerical solution efficiently. The proposed a…
Fractional-order dependent Radial basis functions meshless methods for the integral fractional Laplacian
本文基于分数阶依赖的广义多二次径向基函数推导伪谱公式，提出一种求解积分分数阶拉普拉斯算子及分数阶扩散方程的简单高效无网格方法，并通过数值实验验证了其准确性与效率。
Double fast algorithm for solving time-space fractional diffusion problems with spectral fractional Laplacian
针对含谱分数阶拉普拉斯算子的高维时空分数阶扩散问题，提出结合空间半离散格式与快速时间步进L1格式的双快速算法，证明其稳定性与最优时间收敛阶，并通过数值算例验证有效性。
We consider effective preconditioners for solving Laplacians of general weighted graphs. Theoretically, spectral sparsifiers (SSs) provide preconditioners of optimal computational complexity. However, they are not easy to use for real-world applications due to the implementation complications. Multigrid (MG) methods, on the contrary, are computationally efficient but lack of theoretical justifications. To bridge the gap between theory and practice, we adopt ideas of MG and SS methods and proposed preconditioners that can be used in practice with theoretical guarantees. We expand the original g…
In this paper, we propose a novel method for solving high-dimensional spectral fractional Laplacian equations. Using the Caffarelli-Silvestre extension, the $d$-dimensional spectral fractional equation is reformulated as a regular partial differential equation of dimension $d+1$. We transform the extended equation as a minimal Ritz energy functional problem and search for its minimizer in a special class of deep neural networks. Moreover, based on the approximation property of networks, we establish estimates on the error made by the deep Ritz method. Numerical results are reported to demonstr…
Explicit fractional Laplacians and Riesz potentials of classical functions
本文证明并收集了分数阶拉普拉斯算子及其逆（里斯势）的显式可计算结果，\(f(x)\)有多种选择，还讨论了一些高维情况，旨在为涉及分数阶拉普拉斯算子或里斯势问题的数值方法发展提供帮助。
We propose and analyze a Vanka-type multigrid solver for solving a sequence of complex-shifted Laplacian systems arising in diagonalization-based parallel-in-time algorithms for evolutionary equations. Under suitable assumption, local Fourier analysis shows the proposed Vanka-type smoother achieves a uniform smoothing factor, which is verified by several numerical examples.
In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions (MCFs) as basis functions. As a result, the fractional Laplacian can be fully diagonalised, and the complexity of solving an elliptic fractional PDE is quasi-optimal, i.e., $O((N\log_2N)^d)$ with $N$ being the number of modes in each spatial direction. Ample numerical tests for various decayi…
We propose a novel mathematical framework to address the problem of automatically solving large jigsaw puzzles. This problem assumes a large image, which is cut into equal square pieces that are arbitrarily rotated and shuffled, and asks to recover the original image given the transformed pieces. The main contribution of this work is a method for recovering the rotations of the pieces when both shuffles and rotations are unknown. A major challenge of this procedure is estimating the graph connection Laplacian without the knowledge of shuffles. A careful combination of our proposed method for e…
Scaling Laplacian Pyramids
本文研究了基于拉普拉斯金字塔的Laurent多项式矩阵的可扩展性，提出新的构造方法用于紧波let滤波器组和框架。
Aggregation-based aggressive coarsening with polynomial smoothing
本文提出一种代数多重网格预条件器，通过聚合框架实现激进粗化和多项式平滑，用于图拉普拉斯矩阵系统，展示其简单性和可扩展性。
A novel and simple spectral method for nonlocal PDEs with the fractional Laplacian
本文提出一种基于半离散傅里叶变换的谱方法离散分数阶拉普拉斯算子，具有与其相同的离散符号，可实现谱精度，并借助多水平Toeplitz结构实现快速计算。
Spectral methods for solving elliptic PDEs on unknown manifolds
本文提出一种基于点云数据的无网格谱方法，用于求解未知流形上的椭圆型PDE，采用加权拉普拉斯RBF特征解构造测试空间，结合二阶切空间估计提升精度，理论与实验均验证了收敛性及精度优势。
Stability of Lagrange elements for the mixed Laplacian
通过数值实验研究了拉普拉斯算子混合形式中简单元选择的稳定性，发现对于多项式次数 r=2 或 3，向量拉格朗日元及其散度空间在多种网格族上稳定，而对角网格族上 Stokes 方程不稳定时该组合仍稳定；r=1 时稳定性依赖于网格；收敛性仅出现在稳定方法中，且低次时向量变量的最优阶 L2 估计不成立。
Spectral Method for the Fractional Laplacian in 2D and 3D
本文提出了一种用于近似分数拉普拉斯算子和求解分数泊松问题的谱方法，该方法基于单位球体内分数拉普拉斯算子的特征函数和特征值的显式表达，在加权$ L^2 $空间中具有谱精度和计算效率。
Shifted Chebyshev polynomials for Solving Three-Dimensional Volterra Integral Equations of the second kind
本文提出了一种高效方法，利用移位切比雪夫多项式近似求解具有连续核的三维Volterra积分方程，将积分方程转化为代数方程，并通过Maple 17计算数值结果和误差估计。
An iterative approximate method of solving boundary value problems using dual Bernstein polynomials
本文提出一种新的迭代近似方法，通过最小二乘法结合双伯恩斯坦多项式性质求解线性和非线性微分方程，验证了方法的高效性和广泛适用性。
We introduce a randomized algorithm, namely RCHOL, to construct an approximate Cholesky factorization for a given Laplacian matrix (a.k.a., graph Laplacian). From a graph perspective, the exact Cholesky factorization introduces a clique in the underlying graph after eliminating a row/column. By randomization, RCHOL only retains a sparse subset of the edges in the clique using a random sampling developed by Spielman and Kyng. We prove RCHOL is breakdown-free and apply it to solving large sparse linear systems with symmetric diagonally dominant matrices. In addition, we parallelize RCHOL based o…
Complexity of sparse polynomial solving 3: Infinity
本文提出在环面簇坐标卡上局部定义同伦算法，以求解稀疏多项式系统，避免虚假根，其复杂度由条件长度线性界定，并支持在环面无穷远附近稳定计算。
Decoupled fractional Laplacian wave equation can describe the seismic wave propagation in attenuating media. Fourier pseudospectral implementations, which solve the equation in spatial frequency domain, are the only existing methods for solving the equation. For the earth media with curved boundaries, the pseudospectral methods could be less attractive to handle the irregular computational domains. In the paper, we propose a radial basis function collocation method that can easily tackle the irregular domain problems. Unlike the pseudospectral methods, the proposed method solves the equation i…
Computational complexity of solving polynomial differential equations over unbounded domains with non-rational coefficients
本文在无界域上扩展了Pouly G16关于求解多项式微分方程复杂性的结果，适用于非有理输入，并在可计算分析框架下得到运算符的统一复杂性结果。
Numerical approach for solving problems arising from polynomial analysis
本文提出了一种基于随机根采样的数值方法，用于解决多项式分析中出现的理论问题，该方法在传统方法无法有效解决的情况下表现出色。
In this article, we describe an implementation of a polynomial system solver to compute the approximate solutions of a 0-dimensional polynomial system with finite precision p-adic arithmetic. We also describe an improvement to an algorithm of Caruso, Roe, and Vaccon for calculating the eigenvalues and eigenvectors of a p-adic matrix.
Solving Polynomial Systems in the Cloud with Polynomial Homotopy Continuation
多项式系统出现在科学和工程的许多领域。多项式同伦延拓方法应用符号-数值算法来求解多项式系统。我们描述了我们的Web界面的设计和实现，并反映了多项式同伦延拓方法在云中解决多项式系统的应用。通过图同构问题，我们组织和分类的多项式系统，我们解决。图的标准形式的分类将新提交的系统与已经解决的系统识别出来。
In latest years, several advancements have been made in symbolic-numerical eigenvalue techniques for solving polynomial systems. In this article, we add to this list. We design an algorithm which solves systems with isolated solutions reliably and efficiently. In overdetermined cases, it reduces the task to an eigenvalue problem in a simpler and considerably faster way than in previous methods, and it can outperform the homotopy continuation approach. We provide many examples and an implementation in the proof-of-concept Julia package EigenvalueSolver.jl.
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